For each of these problems you may use either of our two equivalent notions of continuity: sequences or ɛ-d. But be aware that you will be expected to be familiar with both methods. (a) Assume that f: [0, ∞) → R and g : (-∞, 0] are continuous functions such that f(0) = g(0). Prove that the function f: R→R defined by f(x) h(x) = { g(z) I>0 I <0. is continuous with the ɛ-8 method. Then prove it with the sequence method. (b) Prove that there is no continuous function f: R→R such that for all r ER\ {0} we have f(x) = 1.
For each of these problems you may use either of our two equivalent notions of continuity: sequences or ɛ-d. But be aware that you will be expected to be familiar with both methods. (a) Assume that f: [0, ∞) → R and g : (-∞, 0] are continuous functions such that f(0) = g(0). Prove that the function f: R→R defined by f(x) h(x) = { g(z) I>0 I <0. is continuous with the ɛ-8 method. Then prove it with the sequence method. (b) Prove that there is no continuous function f: R→R such that for all r ER\ {0} we have f(x) = 1.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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part a and b
![2. CONTINUOUS FUNCTIONS
For each of these problems you may use either of our two equivalent notions of continuity:
sequences or ε-d. But be aware that you will be expected to be familiar with both methods.
(a) Assume that f: [0,00)→ R and g: (-∞, 0] are continuous functions such that f(0) = g(0).
Prove that the function fƒ: R → R defined by
f(x)
h(x) = { g(7)
x>0
x < 0.
is continuous with the s-d method. Then prove it with the sequence method.
(b) Prove that there is no continuous function f: R→ R such that for all r ER\ {0} we have
f(x) = 1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F04625ac1-43ff-4999-b93a-55388fc0c5e2%2F7c6aeb3f-202a-436b-ae5d-f4995e4566ea%2Fiv4lpg_processed.png&w=3840&q=75)
Transcribed Image Text:2. CONTINUOUS FUNCTIONS
For each of these problems you may use either of our two equivalent notions of continuity:
sequences or ε-d. But be aware that you will be expected to be familiar with both methods.
(a) Assume that f: [0,00)→ R and g: (-∞, 0] are continuous functions such that f(0) = g(0).
Prove that the function fƒ: R → R defined by
f(x)
h(x) = { g(7)
x>0
x < 0.
is continuous with the s-d method. Then prove it with the sequence method.
(b) Prove that there is no continuous function f: R→ R such that for all r ER\ {0} we have
f(x) = 1.
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